Finite Group Databasebeta

← C3×#10592852 #13163188

Subgroups — C3×#10592852 #13163188

Order 1728.

1048 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×D8C211yesmaximal, characteristic
2C3×#64145C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C3×#3754C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7C3×#63641C211yesmaximal, characteristic, complemented
8#10592852C311yesmaximal, characteristic, direct product factor
9C3×#5153—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C2×C3×#634C2211yescharacteristic
14C33×D8—22no—
15C2×C33×C8C2211yescharacteristic
16C3×#3757—21no—
17C3×#3757—21no—
18C3×#3757—21no—
19C3×#3754—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C2×C32×D8—24no—
24C2×C32×D8D314yes—
25C2×C32×D8C2×C311yescharacteristic
26#64145C2×C311yescharacteristic, complemented
27D4×C3⋊Dic3C2×C311yescharacteristic, complemented
28C2×#3754C2×C311yescharacteristic, complemented
29C3×#936—34no—
30C3×D4×Dic3—34no—
31C2×C3×#214—34no—
32#63659C2×C311yescharacteristic, complemented
33#63643C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#832—34no—
36C3×#816—34no—
37C3×Dic3⋊C8—34no—
38C23×C33—21no—
39C23×C33—21no—
40C33×D4—21no—
41C33×D4—21no—
42C33×D4—21no—
43C33×D4D412yes—
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—42no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×#634—21no—
51C3×#634—21no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C3×#463—91no—
55#5153—34no—
56C2×C32×D4—24no—
57C2×C32×D4—24no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C22×C311yescharacteristic
61C2×C32×D4C22×C311yescharacteristic
62C22×C3⋊Dic3—21no—
63C2×#634C22×C311yescharacteristic
64C22×C3×Dic3—64no—
65C2×C3×Dic6—34no—
66C32×D8—216no—
67C32×D8—28no—
68C32×D8—22no—
69C2×C32×C8—24no—
70C2×C32×C8C2×D314yes—
71C2×C32×C8C22×C311yescharacteristic
72#3757—21no—
73#3757—21no—
74#3757—21no—
75#3754—22no—
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic
77C4×C3⋊Dic3C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#217—64no—
80C3×#217—64no—
81C3×#217—64no—
82C3×#214—68no—
83C3×C4⋊Dic3—34no—
84C3×C4×Dic3—34no—
85C2×C3×C3⋊C8—34no—
86C22×C33—22no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33C2×D411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—21no—
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—41no—
98C3×C3⋊Dic3—41no—
99C2×C3×SD16—91no—
100C2×C3×D8—24no—
101C2×C3×D8C3×D314yes—
102C2×C3×D8C3⋊D311yescharacteristic
103C3×#124—91no—
104C3×C4×D4—91no—
105#936—34no—
106D4×Dic3—34no—
107C2×#214—34no—
108C3×C4⋊C8—91no—
109C3×#103—91no—
110C3×#102—91no—
111#832—34no—
112#816—34no—
113Dic3⋊C8—34no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—24no—
117C23×C32—24no—
118C23×C32—44no—
119C23×C32—44no—
120C32×D4—28no—
121C32×D4—24no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—28no—
126C32×D4—24no—
127C32×D4—24no—
128C32×D4—44no—
129C32×D4—44no—
130C32×D4#6718yes—
131C32×D4C3×D412yes—
132C2×C32×C4—24no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C23×C311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—42no—
138C2×C3⋊Dic3C3×D411yescharacteristic
139C2×C3⋊Dic3C3×D411yescharacteristic
140#634—21no—
141#634—21no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—128no—
147C3×Dic6—64no—
148C3×Dic6—64no—
149C32×C8—28no—
150C32×C8—24no—
151C32×C8—21no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—64no—
154#463—91noSylow
155C2×C33—42no—
156C2×C33—81no—
157C2×C33#12111yescharacteristic
158C2×C33#13511yescharacteristic
159C2×C33AGL(1,ℤ/8)11yescharacteristic
160C2×C3×Q8—91no—
161C2×C3×D4—24no—
162C2×C3×D4—24no—
163C2×C3×D4C2×C3×D314yes—
164C2×C3×D4C2×C3×D314yes—
165C2×C3×D4C2×C3⋊D311yescharacteristic
166C2×C3×D4C2×C3⋊D311yescharacteristic
167C22×C3×C4—181no—
168C22×Dic3—64no—
169C2×Dic6—34no—
170C3×SD16—182no—
171C3×D8—216no—
172C3×D8—28no—
173C3×D8—22no—
174C2×C3×C8—24no—
175C2×C3×C8—91no—
176C2×C3×C8C2×C3×D314yes—
177C2×C3×C8C2×C3⋊D311yescharacteristic
178C3×C4⋊C4—91no—
179C3×C22⋊C4—181no—
180C3×C22⋊C4—181no—
181C3×C22⋊C4—181no—
182C3×C42—91no—
183#217—64no—
184#217—64no—
185#217—64no—
186#214—68no—
187C4⋊Dic3—34no—
188C4×Dic3—34no—
189C2×C3⋊C8—34no—
190C22×C32—22no—
191C22×C32—24no—
192C22×C32—28no—
193C22×C32—48no—
194C22×C32—48no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—44no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—44no—
202C22×C32—44no—
203C22×C32—44no—
204C22×C32—44no—
205C22×C32—84no—
206C22×C32—84no—
207C22×C32—84no—
208C22×C32—84no—
209C22×C32D3×D414yes—
210C22×C32C2×C3×D411yescharacteristic
211C32×C4—24no—
212C32×C4—24no—
213C32×C4#23714yes—
214C32×C4C2×#6714yes—
215C32×C4C2×C3×D411yescharacteristic
216C32×C4C3×#4111yescharacteristic
217C3⋊Dic3—21no—
218C3⋊Dic3—21no—
219C3⋊Dic3—41no—
220C3⋊Dic3—41no—
221C3×Dic3—64no—
222C3×Dic3—64no—
223C3×Dic3—124no—
224C3×Dic3—124no—
225C2×SD16—91no—
226C2×D8C3×C3⋊D311yescharacteristic
227#124—91no—
228C4×D4—91no—
229C4⋊C8—91no—
230#103—91no—
231#102—91no—
232C33#46311yescharacteristic, complemented, Sylow
233C23×C3—24no—
234C23×C3—21no—
235C23×C3—24no—
236C23×C3—21no—
237C23×C3—44no—
238C23×C3—44no—
239C3×Q8—181no—
240C3×Q8—181no—
241C3×D4—28no—
242C3×D4—24no—
243C3×D4—24no—
244C3×D4—24no—
245C3×D4—21no—
246C3×D4—21no—
247C3×D4—21no—
248C3×D4—28no—
249C3×D4—44no—
250C3×D4—44no—
251C3×D4C3×#6718yes—
252C3×D4#63812yes—
253C2×C3×C4—24no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4—362no—
259C2×C3×C4C22×C3×D314yes—
260C2×C3×C4C22×C3⋊D311yescharacteristic
261C2×Dic3—34no—
262C2×Dic3—34no—
263C2×Dic3—64no—
264C2×Dic3—64no—
265C2×Dic3—128no—
266Dic6—64no—
267Dic6—64no—
268C3×C8—28no—
269C3×C8—24no—
270C3×C8—21no—
271C3×C8—181no—
272C3⋊C8—64no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32—42no—
277C2×C32—48no—
278C2×C32—88no—
279C2×C32—88no—
280C2×C32—84no—
281C2×C32—81no—
282C2×C32#91114yes—
283C2×C32#91214yes—
284C2×C32#93914yes—
285C2×C32C3×#12111yescharacteristic
286C2×C32C3×#13511yescharacteristic
287C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
288C2×Q8—91no—
289C2×D4C2×C3×C3⋊D311yescharacteristic
290C2×D4C2×C3×C3⋊D311yescharacteristic
291C22×C4—181no—
292SD16—182no—
293D8—22no—
294C2×C8—91no—
295C2×C8C2×C3×C3⋊D311yescharacteristic
296C4⋊C4—91no—
297C22⋊C4—181no—
298C22⋊C4—181no—
299C22⋊C4—181no—
300C42—91no—
301C22×C3—28no—
302C22×C3—22no—
303C22×C3—24no—
304C22×C3—48no—
305C22×C3—48no—
306C22×C3—44no—
307C22×C3—44no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—44no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—84no—
317C22×C3—84no—
318C22×C3—84no—
319C22×C3—84no—
320C22×C3C3×D3×D414yes—
321C22×C3D4×C3⋊D311yescharacteristic
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—181no—
325C3×C4—181no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4C3×#23714yes—
329C3×C4C2×C3×#6714yes—
330C3×C4#383011yescharacteristic
331C3×C4C2×#63811yescharacteristic
332Dic3—64no—
333Dic3—64no—
334Dic3—124no—
335Dic3—124no—
336C32—24no—
337C32#515314yescomplemented
338C32C3×#46311yescharacteristic, complemented
339C23—21no—
340C23—21no—
341Q8—181no—
342Q8—181no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4C3×#63812yes—
347C2×C4—91no—
348C2×C4—91no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—362no—
352C2×C4C22×C3×C3⋊D311yescharacteristic
353C8—21no—
354C8—181no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—42no—
359C2×C3—48no—
360C2×C3—88no—
361C2×C3—88no—
362C2×C3—81no—
363C2×C3—84no—
364C2×C3C3×#91114yes—
365C2×C3C3×#91214yes—
366C2×C3C3×#93914yes—
367C2×C3#6412011yescharacteristic
368C2×C3#6412111yescharacteristic
369C2×C3#6414811yescharacteristic
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22C3×D4×C3⋊D311yescharacteristic
377C4—181no—
378C4—181no—
379C4—361no—
380C4—361no—
381C4C3×#383011yescharacteristic
382C4C2×C3×#63811yescharacteristic
383C3—24no—
384C3C3×#515314yescomplemented
385C3#1059285211yescharacteristic, direct product factor
386C2—42no—
387C2—81no—
388C2C3×#6412011yescharacteristic
389C2C3×#6412111yescharacteristic
390C2C3×#6414811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
Fitting, prime radicalC2×C33×D8C2
socleC22×C33C2×D4
hypercenterC2×C3×D8C3⋊D3
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#64145C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. C2×C3×#3754C2semidirect
5. C3×#63659C2semidirect
6. C3×#63643C2semidirect
7. C3×#63641C2semidirect
8. #10592852C3direct (N × H)
26. #64145C2×C3semidirect
27. D4×C3⋊Dic3C2×C3semidirect
28. C2×#3754C2×C3semidirect
32. #63659C2×C3semidirect
33. #63643C2×C3semidirect
34. #63641C2×C3semidirect
232. C33#463semidirect
337. C32#5153semidirect
338. C32C3×#463semidirect
384. C3C3×#5153semidirect
385. C3#10592852direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C33×D82
C3×#641452
C3×D4×C3⋊Dic32
C2×C3×#37542
C3×#636592
C3×#636432
C3×#636412
#105928523
C3×#51533